Adam opened the scoring for the Reds from the spot after Luis Suarez was alleged to have been brought down in the penalty area.
Carroll then extended his side's lead just before the break after latching unto a through ball by Suarez before sliding the ball past Ben Foster.
Speaking after the game, Adam was full of praise for Carroll as he expressed his delight with the victory.
"The big man has been unbelievable today [Saturday] and the finish for our second goal was terrific," he told the club's official website.
"Once we got the goal it changed the complexion of the game. I thought we played well in the first half. Second half, we never kept the ball as well as we should have and it's something that we'll maybe have to get better at.
"But to come here and get three points has delighted us.
"We're disappointed with some of the results we've had this year. Last week (v Norwich) we had numerous chances and could have finished it off but that's what happens when you play in the best league in the world. If you don't score you can get punished, and we did.
"We've done that a couple of times this year but today [Saturday] we've started the game very well and deserved to win the game in the end."
Liverpool are sixth in the Barclays Premier League table and host Swansea on Saturday.